Anomalous Resonance Phenomena of Solitary Waves with Internal Modes

VOLUME 84, NUMBER 5 PHYSICAL REVIEW LETTERS 31JANUARY 2000 Anomalous Resonance Phenomena o Soli a y Wa es wi h In e nal Modes Niu ka R. Quin e o* and Angel Sánchez† G upo In e disciplina de Sis emas Complicados (GISC), Depa amen o de Ma emá icas, Uni e sidad Ca los III de Mad id, A enida de la Uni e sidad 30, E-28911 Leganés, Mad id, Spain F anz G. Me ens‡ Physikalisches Ins i u , Uni e si ä Bay eu h, D-95440 Bay eu h, Ge many (Recei ed 16 July 1999) We in es iga e he nonpa ame ic, pu e ac d i en dynamics o nonlinea Klein-Go don soli a y wa es ha ing an in e nal mode o equency Vi. We show ha he s onges esonance a ises when he d i ing equency d苷Vi兾2, whe eas when d苷Vi he esonance is weake , disappea ing o nonze o damping. A esonance, he dynamics o he kink cen e o mass becomes chao ic. As we iden i y he esonance mechanism as an indi ec coupling o he in e nal mode due o i s symme y, we expec simila esul s o o he sys ems. PACS numbe s: 05.45.Y , 02.30.J , 03.50.–z, 63.20.Pw An impo an pa adigm es ablished o e he las wo decades is ha soli a y wa es o soli ons beha e e y much like poin pa icles when subjec ed o (a la ge class o ) ex- e nal o ces and pe u ba ions [1–3]. Howe e , many soli- a y wa es possess one (some imes mo e han one) in e nal o shape mode [4,5], and in ha case he pa icle pic u e o hei dynamics may be o e simpli ied: Indeed, he in e - nal mode can empo a ily s o e ene gy and elease i a a la e s age, gi ing ise o esonance phenomena in soli a y wa e collisions [5] o in soli a y wa e in e ac ions wi h in- homogenei ies [6]. As in e nal modes a e qui e common in nonlinea sys ems, ei he in insically o as a esul o small pe u ba ions [7], he ques ion o hei in luence on he dynamics o soli a y wa es is a e y gene al and ele- an one. One aspec o soli a y wa e dynamics ha has p o en i sel di icul o unde s and is ha o opological soli a y wa es o kinks subjec ed o pu e, i.e., nonpa ame ic ac d i ing. Thus, only ecen ly [8] he ac d i en dynamics o sine-Go don kinks ( ha do no possess an in e nal mode) has been de ini ely cla i ied. Nai ely, he only new phe- nomenon one expec s when a nonpa ame ic ex e nal d i - ing ac s on soli ons wi h in e nal modes is a esonance when i s equency, D, ma ches ha o an in e nal mode, Vi. The aim o his Le e is o show ha , in ac , he ac ual scena io is mos unexpec ed and highly non i ial. As we will see below, a s ong, anomalous esonance a ises when d苷Vi兾2, whe eas he no mal esonance a d苷Viis de ini ely weake , only possible a exac ly ze o damping, and e en hen i can be supp essed by app op ia e choices o o he pa ame e s. We expec his esul o be gene ic, because ou analy ical app oach allows us o iden i y he mechanism o such a peculia phenomenon: The ac o ce does no ac di ec ly on he in e nal mode (because o sym- me y easons), bu a he , hey in e ac indi ec ly ia he ansla ional mo ion which couples o he in e nal mode. Ou p edic ions a e ully con i med by nume ical simu- la ions, which in addi ion show he implica ions o hese esonances o he kink dynamics. As a speci ic example o a kink wi h in e nal mode, we ake he well known [1] 4equa ion, which, when d i en wi h an ac o ce 共 兲苷esin共d 1d 0兲, eads 2 xx 1U0共 兲苷2b 1 共 兲,(1) whe e U共 兲苷共 221兲2兾4and bis a damping coe i- cien . P e ious ela ed wo ks on his sys em a e [9], whe e esonances in he p esence o an ex e nal ( ime indepen- den ) po en ial ha e been conside ed, and [10], which deal wi h non esonan , high equency pa ame ic ac d i ings. Fo ou p oblem, ou analy ical app oach will be he well known collec i e coo dina e (CC) me hod [2,3]. A i s o de o app oxima ion is gi en by he McLaughlin-Sco me hod [11]: We assume ha he solu ion o (1) is o he o m 共x, 兲苷 anh"x2X共 兲 l0p12V共 兲2#,(2) whe e l0苷p2. The cen e o he kink X共 兲and i s e- loci y V共 兲a e ela ed by X共 兲苷R 0d 0V共 0兲1X共0兲, and bo h a e unknown unc ions desc ibing he mo ion o he kink as a cohe en en i y. By means o a s anda d p oce- du e [2,3,8,11] in ol ing conse a ion laws, an o dina y di e en ial equa ion o mo ion o V共 兲can be ob ained, linea ized [8], and sol ed, yielding V共 兲苷 共 兲 p11 共 兲2,(3) 共 兲⬅¯ce2b 23p2e 2共b21d 2兲 3关bsin共d 1d 0兲2dcos共d 1d 0兲兴 ,(4) ¯c苷g0V共0兲13p2e 2共b21d 2兲关bsin共d0兲2dcos共d0兲兴 , 0031-9007兾00兾84(5)兾871(4)$15.00 © 2000 The Ame ican Physical Socie y 871 VOLUME 84, NUMBER 5 PHYSICAL REVIEW LETTERS 31J ANUARY 2000 whe e g0⬅1兾p12V共0兲2. Fo he undamped case (b苷0), we see ha he kink oscilla es i g0V共0兲苷 3p2ecos共d0兲兾共2d兲; o he wise, he kink will mo e ei he o he igh o o he le , depending on he ela ion be ween he pa ame e s o he ac o ce and he ini ial eloci y. This dc mo ion is absen when bfi0, as has been nume ically con i med o sine-Go don kinks in [8]. In o de o include in e nal mode e ec s, we p oceed as ollows: We ew i e Eq. (1) as ᠨ c苷2dH d 2bᠨ 1 共 兲,ᠨ 苷dH dc ,(5) whe e c苷ᠨ , he do meaning de i a i e wi h espec o ime, and H苷Z1` 2` dx Ω1 2c211 2 2 x1U共 兲æ(6) is he Hamil onian o he sys em when e苷b苷0. We now make he ansa z 共x, 兲苷 关x2X共 兲,l共 兲兴, whe eas om he de ini ion o cwe ha e ha c共x, 兲苷 c关x2X共 兲,l共 兲,ᠨ X,ᠨ l兴. As in he McLaughlin-Sco me hod, X共 兲 ep esen s he kink cen e posi ion, bu now we in oduce a second collec i e a iable l共 兲 ha will s and o he kink wid h, i.e., he in e nal mode exci a ion, below. The p ocedu e o ob ain he CC equa ions co espond- ing o his gene alized a eling wa e ansa z has been pu o wa d in [12,13]. Basically, i consis s o inse ing ou ansa z in o (5), mul iplying he i s equa ion by ≠ 兾≠X and he second one by ≠c兾≠X(≠ 兾≠land ≠c兾≠l), ak- ing hei di e ence, and in eg a ing o e x. This yields a a he cumbe some, o dina y di e en ial equa ion o X共 兲 [l共 兲], which we omi he e o b e i y. The nex s ep is o choose a speci ic unc ional o m o , which we do ollowing he wo k o Rice [14], and le 关x2X共 兲,l共 兲兴 苷 0∑x2X共 兲 l共 兲∏,(7) whe e 0关共x2X兲兾l0兴is he s a ic kink solu ion, which is an odd unc ion (wi h espec o i s cen e ) o he 4and o any o he e en po en ial. Upon pa icula iza ion o he CC equa ions o his o m o we inally ob ain M0l0 ¨ X l2M0l0 ᠨ Xᠨ l l2 苷2bM0l0 ᠨ X l22 共 兲,(8) aM0l0 ¨ l l1M0l0 ᠨ X2 l2 苷Kin 共l,ᠨ l,ᠨ X兲2 baM0l0 ᠨ l l; (9) whe e Kin 苷2≠E兾≠land E苷1 2 l0 lM0 ᠨ X21l0 2laM0 ᠨ l211 2M0µl0 l1l l0∂ (10) is he kink ene gy. Impo an ly, in ob aining Eq. (9) a e m o he o m 共 兲R` 2` dx ≠ 0兾≠l, coming om he cou- pling o he ac d i ing o he in e nal mode, has anished because o symme y. Fo he 4equa ion, 0苷 anhxin Eq. (7), which yields a苷共p226兲兾12 and M0苷2p2兾3. Le us now simpli y hese exp essions in o de o make i s physical signi icance mo e anspa en . To begin wi h, Eq. (8) can be sol ed o ᠨ X兾l, yielding P⬅M0l0 ᠨ X l 苷22ebsin共d 1d 0兲2dcos共d 1d 0兲 共b21d 2兲 1e2b ∑2ebsin共d0兲2dcos共d0兲 共b21d 2兲1M0l0V共0兲 ls∏, (11) whe e ls苷l0兾g0. Inse ing Eq. (11) in o Eq. (9), we ind a关ᠨ l222l¨ l22blᠨ l兴苷l2 l2 0∑11P2 M2 0∏21. (12) Al hough Eqs. (11) and (12) a e qui e complica ed, choos- ing V共0兲and he phase d0so ha he exponen ial e ms in Eq. (11) anish, we can use a change o a iables (p o- posed in [15] o he dc d i ing case) o ans o m he Eq. (12) in o a Pinney-like equa ion (see [16] and e e - ences he ein), which can be sol ed in e ms o Ma hieu unc ions [17] i b苷0(when bfi0we ha e no been able o sol e his p oblem analy ically). In any e en , we do no need he analy ical exp essions o he solu ion o he sys em (11) and (12) o unde s and he physics p edic ed by ou app oach. The e m P2in Eq. (12) is an oscilla o y unc ion wi h equency 2d[see Eq. (11)]; hence, we can immedia ely expec a esonance when he ex e nal equency dis hal he equency o he in e nal mode, VR苷1兾pal0in he Rice app oxima ion. Fo he 4model VRo e es ima es Vi苷p3兾2by 1.7% [14]. The analy ical solu ion o b苷0con i ms his ex- pec a ion, while nume ical in eg a ion o Eq. (12) p o es ha he beha io o lis ha o a esonan , damped oscilla- o when bfi0. When d苷VR, inspec ion o Eqs. (11) and (12) leads o he conclusion ha ano he esonance should be ound a d苷VRonly i b苷0(o he wise i is a ansien phenomenon o li e ime b21); e en hen, by choosing V共0兲and d0 o cancel he nonoscilla o y e ms in Eq. (11) he esonance is comple ely supp essed. Fu he - mo e, bo h analy ically and nume ically we ha e e i ied ha , a away om he esonances, he beha io o he kink cen e , X共 兲, is p ac ically he same as he one p edic ed by he McLaughlin-Sco app oach, Eq. (3). This means ha , wi hin he CC amewo k, he beha io o ac d i en 4 kinks is desc ibed by he McLaughlin-Sco ansa z, and only o d i ings close o VR兾2(and VRi b苷0) such app oach ails and esonan phenomena a ise. A his poin , wo key issues mus be add essed: Fi s , unde lying CC me hods is he assump ion ha no (o a negligible amoun o ) adia ion is gene a ed by he pe u - ba ion, an assump ion whose alidi y can be assessed only h ough compa ison wi h he co esponding PDE. Sec- ond, e en i ha is he case, he CC equa ions p edic 872 VOLUME 84, NUMBER 5 PHYSICAL REVIEW LETTERS 31J ANUARY 2000 an unbounded g ow h o l共 兲a esonance, and i is di - icul o unde s and wha ha means in physical e ms o he kink o he ull PDE, whose wid h is con olled by he p ope ies o he equa ion. In iew o his, we ha e compu ed he nume ical solu ion o he PDE (1) by using he conse a i e S auss-Vázquez scheme [18] wi h Dx苷0.1,D 苷0.01, and a o al sys em leng h o L苷400. Ou ini ial condi ion was a kink a es and d0苷p兾2, a pai o alues o which we should no see a esonance a Vi. We come back o his below. We moni- o ed he posi ion and he eloci y o he kink cen e as well as he o al ene gy in he sys em, compu ed om he Hamil onian (6). We also ied o measu e di ec ly he kink wid h, bu we ound ha i is qui e complica ed o es ima e i om he nume ics, his being he eason why we ha e eso ed o less di ec measu emen s. Figu e 1 shows examples o he kink cen e dynamics and i s ene gy e olu ion bo h close o and away om he p edic ed esonance. O - esonance, he beha io o bo h magni udes is pe iodic, whe eas a he esonance i be- comes chao ic. Speci ically, he ene gy inc eases wi h ime: The close o he esonance alue, he as e he inc emen . The cen e mo ion is ini ially pe iodic, un il he in e nal mode ampli ude has inc eased oo much and s o ed oo much ene gy, subsequen ly eleasing i h ough i s coupling wi h he ansla ion mode (which we know ex- is s om [5]), e en ually yielding he kink mo ion e a ic as his p ocess is epea ed once and again. This is a clea e idence in a o o he esonance p edic ed om he CC ea men . We ha e e i ied ha a esonance he kink mo- ion is e y sensi i e o changes in he ini ial condi ions, hence ou claim o he appea ance o chaos. Figu e 2 depic s he esonance as seen h ough he mean ene gy, compu ed as a ime a e age om 苷10 000 (a - e ansien s ha e died ou in he damped case) o he end o he un a 苷25 000. Figu e 2(a) shows he beha io o his magni ude a ound Vi兾2. I is clea om he plo ha he e is a s ong esonance a d艐0.6102 艐Vi兾2, bo h wi h and wi hou dissipa ion. In ac , some poin s a e missing in he b苷0line because he co esponding kinks, in hei appa en ly andom mo ion, le he sys em be o e he end o he un. On he o he hand, in spi e o ou choice o ini ial condi ions, o which no esonance is p e- dic ed a Vi o b苷0, a weak esonance can be seen a d艐1.225 艐Viin Fig. 2(b). We no e ha he o he wo small peaks a e spu ious, since hey appea , disappea , o change loca ion depending on he choice o he leng h o he nume ical simula ion. We belie e ha his disc epancy migh come om he di icul y o nume ically uning he condi ion o i s supp ession. In any e en , ha would be a e y special case, and he beha io we ind o hese pa- ame e s is ep esen a i e o wha occu s o o he choices o he ini ial eloci y and he phase. Those gi e ise o a simila beha io , while, ema kably, he esonance a Viis always weake and na owe han ha a Vi兾2(c . Fig. 2). We also see ha he p edic ion ha he esonance a Viis supp essed by he dissipa ion is also con i med, he small 0 5000 10000 15000 20000 25000 0.9 1.1 1.3 1.5 1.7 Ene gy 0 2500 5000 0.9 1.0 1.1 1.2 Ene gy (a) 0 5000 10000 15000 20000 25000 −100 −60 −20 20 60 100 X( ) 0 2500 5000 −20 −10 0 10 X( ) (b) FIG. 1. Resul s o he PDE (1) wi h b苷0,e苷0.01. (a) To al ene gy when d苷0.6100 (uppe line), d苷0.6080 (lowe line). (b) Same o he kink cen e , X共 兲. Inse s show he same o d苷0.6102, a alue o which he kink eaches he bounda y o he nume ical sys em be o e 苷25 000. peak in he lowe line s emming om he ansien s, which a e no exac ly ze o o 10 000 # #25 000. Ano he in- e es ing ema k is ha Fou ie analysis shows ha he mo- no onous inc easing o he ene gy ha appea s in Fig. 2(b) comes om he ac ha , when d*1.1, he lowes phonon mode (wi h equency p苷p2) begins o be exci ed, he ampli ude o i s exci a ion mono onically inc easing as d!p2. No e idence o lowes phonon mode exci a ion is seen o he esonance a Vi兾2; he e o e, his is indeed a phenomenon a ising om he coupling o he ansla ion and he in e nal mode as p edic ed by he CC calcula ion. In summa y, we ha e s udied how ac o ces a ec soli a y wa es o kink ype wi h an in e nal mode o equency Vi. Speci ically, we ha e clea ly shown ha he beha io o 4kinks unde ac d i ing is e y well desc ibed by he wo- a iable CC heo y we ha e de eloped he e. The main ea u e o he nonpa ame i- cally, ac d i en 4kink dynamics is ha he s onges esonance occu s a d苷Vi兾2, and no a he equency one would expec , d苷Vi. We emphasize ha his no el esonance phenomenon is o ally unexpec ed om he knowledge o he in e nal mode equency, and a ises 873 VOLUME 84, NUMBER 5 PHYSICAL REVIEW LETTERS 31J ANUARY 2000 0.59 0.60 0.61 0.62 0.63 δ 0.92 1.00 1.08 1.16 1.24 1.32 1.40 Mean Ene gy (a) 1.21 1.22 1.23 1.24 1.25 1.26 δ 0.98 1.08 1.18 1.28 1.38 Mean Ene gy (b) FIG. 2. Resul s o he PDE (1). Mean alue o he ene gy (see ex o he way i is compu ed) s d i ing equency d, (a) close o Vi兾2, (b) close o Vi. In bo h cases, he uppe line co esponds o b苷0, and he lowe line o b苷0.001. om he indi ec in e ac ion o he ex e nal o ce wi h he in e nal mode ia he ansla ional mo ion. Al hough ou esul s ha e been ob ained o a speci ic example, he 4 equa ion (1), o he models wi h e en on-si e po en ials and in e nal modes, such as he double sine-Go don equa ion, o ins ance, will beha e simila ly because a CC app oach will lead o analogous esul s. The esonance ound a he CC le el mani es s i sel a he PDE le el as e a ic o chao ic (s ongly dependen on he ini ial condi ions) mo ion o he kink as he kine ic ene gy o he cen e o mass is s o ed in o, and eco e ed om, he in e nal mode. To conclude, we no e ha he anomalous esonances de- sc ibed he e a e impo an on hei own, as examples o he highly non i ial beha io o nonlinea sys ems and as hin s abou he mechanisms go e ning kink dynamics. In addi ion, we hink ha his phenomenon should be e y gene al, in iew o he ecen inding [7] ha pe u ba- ions o kink-bea ing nonlinea sys ems o en lead o he de elopmen o an in e nal mode. As his occu s in he disc e e sine-Go don model [7], a esonance like he one discussed he e could be ele an o he mode locking phe- nomena epo ed o ha sys em in [19]. Finally, by using his disc e eness induced in e nal mode, we poin ou ha he esonance we ind could be obse ed in expe imen s by using Josephson junc ion a ays as in [20]. We hank Yu i Gaididei, F ancisco Domínguez-Adame, and JoséCues a o discussions. Wo k a GISC (Leganés) has been suppo ed by DGESIC (Spain) G an No. PB96- 0119. T a el be ween Bay eu h and Mad id has been suppo ed by “Acciones In eg adas Hispano-Alemanas,” a join p og am o DAAD (Az. 314-AI) and DGESIC. *Elec onic add ess: [email p o ec ed] †Elec onic add ess: [email p o ec ed] ‡Elec onic add ess: [email p o ec ed].uni-bay eu h.de [1] A.C. 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